Optimal. Leaf size=48 \[ -\frac {a^2}{b^3 n \left (a+b x^n\right )}-\frac {2 a \log \left (a+b x^n\right )}{b^3 n}+\frac {x^n}{b^2 n} \]
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Rubi [A] time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {266, 43} \[ -\frac {a^2}{b^3 n \left (a+b x^n\right )}-\frac {2 a \log \left (a+b x^n\right )}{b^3 n}+\frac {x^n}{b^2 n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{-1+3 n}}{\left (a+b x^n\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{(a+b x)^2} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{b^2}+\frac {a^2}{b^2 (a+b x)^2}-\frac {2 a}{b^2 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {x^n}{b^2 n}-\frac {a^2}{b^3 n \left (a+b x^n\right )}-\frac {2 a \log \left (a+b x^n\right )}{b^3 n}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 38, normalized size = 0.79 \[ \frac {-\frac {a^2}{a+b x^n}-2 a \log \left (a+b x^n\right )+b x^n}{b^3 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 59, normalized size = 1.23 \[ \frac {b^{2} x^{2 \, n} + a b x^{n} - a^{2} - 2 \, {\left (a b x^{n} + a^{2}\right )} \log \left (b x^{n} + a\right )}{b^{4} n x^{n} + a b^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3 \, n - 1}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 59, normalized size = 1.23 \[ -\frac {2 a \ln \left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )}{b^{3} n}+\frac {\frac {{\mathrm e}^{2 n \ln \relax (x )}}{b n}-\frac {2 a^{2}}{b^{3} n}}{b \,{\mathrm e}^{n \ln \relax (x )}+a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 61, normalized size = 1.27 \[ \frac {b^{2} x^{2 \, n} + a b x^{n} - a^{2}}{b^{4} n x^{n} + a b^{3} n} - \frac {2 \, a \log \left (\frac {b x^{n} + a}{b}\right )}{b^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^{3\,n-1}}{{\left (a+b\,x^n\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 157.50, size = 129, normalized size = 2.69 \[ \begin {cases} \frac {\log {\relax (x )}}{a^{2}} & \text {for}\: b = 0 \wedge n = 0 \\\frac {x^{3 n}}{3 a^{2} n} & \text {for}\: b = 0 \\\frac {\log {\relax (x )}}{\left (a + b\right )^{2}} & \text {for}\: n = 0 \\- \frac {2 a^{2} \log {\left (\frac {a}{b} + x^{n} \right )}}{a b^{3} n + b^{4} n x^{n}} - \frac {2 a^{2}}{a b^{3} n + b^{4} n x^{n}} - \frac {2 a b x^{n} \log {\left (\frac {a}{b} + x^{n} \right )}}{a b^{3} n + b^{4} n x^{n}} + \frac {b^{2} x^{2 n}}{a b^{3} n + b^{4} n x^{n}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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